Rank‐width is less than or equal to branch‐width

Oum, Sang‐il

doi:10.1002/jgt.20280

Cited by 43 publications

(32 citation statements)

References 10 publications

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“…Intuitively speaking, many structured graphs have bounded rank-width. E.g., highly sparse random graphs have a bounded rank-width [40], and so do graphs with a bounded tree-width [41]. For those graphs, the above observation readily applies, and it can be decided whether resource states can be extracted.…”

Section: Proposition 1 (No Bottleneck)mentioning

confidence: 98%

Quantum network routing and local complementation

2019

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Quantum communication between distant parties is based on suitable instances of shared entanglement. For efficiency reasons, in an anticipated quantum network beyond point-to-point communication, it is preferable that many parties can communicate simultaneously over the underlying infrastructure; however, bottlenecks in the network may cause delays. Sharing of multi-partite entangled states between parties offers a solution, allowing for parallel quantum communication. Specifically for the two-pair problem, the butterfly network provides the first instance of such an advantage in a bottleneck scenario. The underlying method differs from standard repeater network approaches in that it uses a graph state instead of maximally entangled pairs to achieve longdistance simultaneous communication. We will demonstrate how graph theoretic tools, and specifically local complementation, help decrease the number of required measurements compared to usual methods applied in repeater schemes. We will examine other examples of network architectures, where deploying local complementation techniques provides an advantage. We will finally consider the problem of extracting graph states for quantum communication via local Clifford operations and Pauli measurements, and discuss that while the general problem is known to be NP-complete, interestingly, for specific classes of structured resources, polynomial time algorithms can be identified. IntroductionQuantum communication schemes over optical networks necessarily suffer from transmission losses and errors. For this reason, in order to achieve the vision of secure quantum communication over arbitrary distances, several schemes have been proposed that are based on entanglement swapping and purification [1][2][3][4][5]. However, such existing "quantum repeater" approaches are based on sharing and manipulating close to maximally entangled "EPR" pairs between the nodes. A lot of emphasis has been put onto identifying efficient ways of achieving this task [2, 3, 6, 7], amounting to challenging prescriptions. Yet, for multi-partite quantum networks going beyond point-to-point achitectures, much less is known about how to meaningfully make use of and manipulate resources. This is particularly unfortunate since a number of protocols have been devised for tasks like secret sharing [8,9], quantum voting [10] and quantum conference key agreement [11][12][13], that exploit the genuine multi-partite character of a quantum network, having the vision of a quantum internet in mind [14]. In fact, one could argue that the true potential of quantum communication is expected to lie in such multi-partite applications beyond point-to-point architectures.Specifically in multi-partite quantum networks, it could well be preferable that the involved processes are run offline, i.e., before a request for communication is received. However, methods like the ones described in Ref. [15] require big quantum memories, as well as a high channel capacity. Consequently, network efficiency is limited by the memory capa...

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Section: Proposition 1 (No Bottleneck)mentioning

confidence: 98%

Quantum network routing and local complementation

2019

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“…We will denote the tree‐width of a graph G as tw ( G ). The following inequality was proved by Oum : for every graph G , we have

…”

Section: Introductionmentioning

confidence: 99%

Rank‐width of random graphs

Lee

Oum

2011

Journal of Graph Theory

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Rank-width of a graph G, denoted by rw(G), is a width parameter of graphs introduced by Oum and Seymour (2006). We investigate the asymptotic behavior of rank-width of a random graph G(n, p). We show that, asymptotically almost surely,iii) if p = c/n and c > 1, then rw(G(n, p)) ≥ rn for some r = r(c), and (iv) if p ≤ c/n and c < 1, then rw(G(n, p)) ≤ 2. As a corollary, we deduce that G(n, p) has linear tree-width whenever p = c/n for each c > 1, answering a question of Gao (2006).

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“…Using this, the first statement follows from Lemmata 4 and 6. The second statement follows from the first by using a theorem from [19] stating that rw(I(G)) ∈ {bw(G), bw(G) − 1}.…”

Section: No Two Edges Of G Are Adjacent)mentioning

confidence: 99%

“…We prove for every graph G with bw(G) = 0 that boolw(G) ≤ bw(G). For the proof we develop a general method of constructive manipulations of the decompositions, which also allows a simplified proof of a theorem by Oum [19] showing that rw(G) ≤ bw(G) (unless E(G) = ∅ and no two edges of G are adjacent). While Oum's proof uses deep results from matroid theory, our argument avoids matroid theory and is based on constructive manipulations of the decompositions, giving a good understanding of the connections between the graph parameters.…”

Section: We See That If Anymentioning

confidence: 99%

On the Boolean-Width of a Graph: Structure and Applications

Adler¹,

Bui-Xuan²,

Rabinovich³

et al. 2010

Graph Theoretic Concepts in Computer Science

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Boolean-width is a recently introduced graph invariant. Similar to tree-width, it measures the structural complexity of graphs. Given any graph G and a decomposition of G of booleanwidth k , we give algorithms solving a large class of vertex subset and vertex partitioning problems in time O * (2 O(k 2 ) ) . We relate the boolean-width of a graph to its branch-width and to the booleanwidth of its incidence graph. For this we use a constructive proof method that also allows much simpler proofs of similar results on rank-width by Oum (JGT 2008). For a random graph on n vertices we show that almost surely its boolean-width is Θ(log 2 n) -setting boolean-width apart from other graph invariants -and it is easy to find a decomposition witnessing this. Combining our results gives algorithms that on input a random graph on n vertices will solve a large class of vertex subset and vertex partitioning problems in quasi-polynomial time O * (2 O(log 4 n) ) .

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Rank‐width is less than or equal to branch‐width

Cited by 43 publications

References 10 publications

Quantum network routing and local complementation

Quantum network routing and local complementation

Rank‐width of random graphs

On the Boolean-Width of a Graph: Structure and Applications

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